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This paper proposes a novel genetic algorithm-based systematic reasoning approach using an orthogonal array crossover (OAX) for solving the traveling salesman problem (TSP). OAX makes use of the systematic reasoning ability of orthogonal arrays that can effectively preserve superior sub-paths from parents and guide the solution towards better quality. OAX combines the advantages of two traditional approaches: canonical approach and heuristic approach. It is shown empirically that OAX outperforms various superior crossovers in both accuracy and speed. An improved OAX with a well-known heuristic method is also presented.